Network-based synchronization of T–S fuzzy chaotic systems with asynchronous samplings
Introduction
Because of different properties of chaotic systems, which include sensitivity toward the initial conditions and system parameters, no periodicity, and topological transitivity, and pseudo-random property, chaotic systems have been explored by various researchers for several decades. The synchronization phenomenon, which is to use the output of the master (or drive) system to control the slave (or response) system so that they oscillate in a synchronized manner, is one of the important topics in the field of chaotic systems and nonlinear dynamics. Indeed, chaotic systems have attracted increasing attention to study the relevant theories and applications, since its inception by Carroll and Pecora [1]. Applications of chaotic systems can be found in different areas [2], [3], [4], such as spread spectrum waveforms, secure communication, image encryption, and cryptography. Various control methods have been employed for chaos synchronization such as adaptive control [5], observer-based control [6], sliding mode control [7], dynamic control [8], and sampled-data control [9], [10], [11].
In recent studies on chaotic synchronization, chaotic master and slave systems are connected directly. With advancement in communication technology, many systems/plants in real industries are now connected through a network, which brings a new synchronization concept known as network-based synchronization. Compared with traditional synchronization scheme, network-based synchronization gives many advantages, such as low cost, reduced weight, simple installation, easy maintenance, and flexible system structure. In network-based synchronization schemes, there are many influencing factors, e.g. external disturbance and sampling effect, which could degrade the performance of the controller, or even disturb the stability of the system. In the real-world environment, the network quality is not always ideal, owing to abrupt changes of network circumstances or customer requests. As such, it is unavoidable that signals transmitted through a network are affected by external disturbances Moreover, in order to send continuous-time signals through a network, the signal must be sampled and encoded in a digital format by a network device (or sensor). Then, the controller has to compute the control inputs from the received sampled-data information, and the plant also should be controlled by sampled-data control inputs. As it is well-known, sampled-data signals are kept constant during the sampling period, and are allowed to change at the sampling instants. These discontinuous-time signals make the analysis difficult and challenging. Therefore, it is necessary to consider sampling effects and disturbances in network-based synchronization schemes. To date, the network-based concept has been widely employed for control of linear/nonlinear systems [12], [13], [14], [15], [16], [17], [18], [19], [20]; however, there are only a few papers on network-based synchronization of chaotic systems, despite its importance. In [21], the problem of network-based master-slave synchronization for delayed chaotic neural networks with network-induced delays, packet dropouts, and stochastic fluctuations was investigated. A logic data processor and a logic zero order hold were proposed for dealing with asynchronous samplings.
Recently, fuzzy-based models have been recognized as a powerful approach to handling nonlinear systems. Among various fuzzy models, the Takagi–Sugeno (T–S) fuzzy model has been widely used for design and analysis of fuzzy control systems. In this regard, the dynamics of chaotic systems can be represented as a weighted sum of some linear sub-systems with simple T–S fuzzy rules [22], [23]. This structure (combination of linear sub-systems by fuzzy rules) is favorable for system analysis and controller design because of its simplicity in mathematical representations. Therefore, T–S fuzzy models have been successfully used in many studies in the literature [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35]. In [30], the problem of fault-tolerant dissipativity-based synchronization for a T–S fuzzy chaotic system with external disturbances was investigated. A new mixed fuzzy delayed feedback dissipativity-based synchronization strategy was proposed. Based on the Ge-Yao-Chen partial region stability theory, a novel adaptive control strategy to increase the efficiency of adaptive control was proposed in [32]. In [33], an adaptive synchronization of T–S fuzzy chaotic systems with uncertain and mismatched parameters was studied. An adaptive observer was designed for estimating the uncertain or mismatched parameters.
Motivated by the success of T–S fuzzy models as described in the above discussion, network-based synchronization of fuzzy chaotic systems with asynchronous samplings and disturbances are investigated in this paper. The main considerations of this paper are as follows:
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The asynchronous samplings between master system, slave system, and controller
The controller receives sampled state information of the master and slave systems, but the signals are sampled at different sampling instants. Therefore, it is difficult to compute suitable control signals by using these asynchronously sampled signals. In order to process the signals, an observer is designed for estimating the states of the master system. Then, we use the state information of the observer, instead of the master one, by sampling using the same sampling instants as those of the received sampled-data of the slave system. The controller produces the control input by using sampled-data of the slave system as well as the observer. So, the computed control input is sampled-data as well. In order to apply the control input to the slave system through a network, the sampled-data control input has to be sampled again with different samplings. As a result, the time relationship among original states, computed control input, and final applied control input is very complicated. In this regard, we derive the time relationship between the signals and formulate two novel models to process the signals.
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The mismatch between fuzzy rules
Unlike many previous investigations on synchronization of fuzzy chaotic systems [29], [30], [31], [32] that assume the same fuzzy rules in both master and slave systems, we consider different fuzzy rules for each fuzzy system. To date, chaos synchronization with mismatched fuzzy rules have been investigated only in a few papers in the literature [33], [34], [35]. As such, this problem is still an open area.
Based on the two established models, the problem of designing an observer and controllers is formulated in terms of linear matrix inequalities(LMIs) by using convex combination techniques and taking the discontinuous type Lyapunov functional approach which fully uses the information of the sampling patterns. Finally, a numerical example with simulation results is presented to demonstrate the effectiveness of the proposed methods.
Notations: is the n-dimensional Euclidean space, X > 0 (respectively, X ≥ 0) means that matrix X is a real symmetric positive definite matrix (respectively, positive semi-definite). I denotes the identity matrix with appropriate dimensions. ⋆ in a matrix represents the elements below the main diagonal of a symmetric matrix. Sym{X} indicates . quot(a, b) indicates the quotient function, and its output is . denotes the remainder of the Euclidean division of a by b. means that the elements of matrix X include the values of f(t). The space of functions which are absolutely continuous on [a, b), have a finite and have square integrable first order derivatives is denoted by Wn[a, b) with the norm .
Section snippets
Problem statement
Consider the following fuzzy system:
Master system rule i: IF z1(t) is Mi1 and and zp(t) is Mip THEN where is a vector of the premise variables, Mij () are fuzzy sets, r is the number of fuzzy rules, is the state vector of the chaotic master system, and is a known constant matrix.
Using the singleton fuzzifier, product fuzzy inference, and weighted average defuzzifier, the fuzzy chaotic master
Observer design
In this section, the observer design is investigated for estimating the states of the master system (2).
The fuzzy observer is described by the following rules.
Observer rule i: IF is Mi1 and and is Mip THEN where is a vector of the premise variables, is the state vector of the observer, and Li is the observer gain matrix to be designed later.
Then,
Controller design
To process asynchronous samplings between the measurement of both master and slave systems, an observer for the master system in the controller is designed in the previous section. The observer states can monitor the original ones with sufficiently small error. And then, using estimated states we can produce . By using this result, we consider the following fuzzy controller:
Controller rule i: IF is Mi1 and and is Mip THEN
Numerical example
In this section, a numerical example with simulation is given to illustrate the validity and superiority of the proposed scheme.
As discovered in many literature [22], [23], we consider the fuzzy chaotic Lorenz master and slave systems in the form of Eqs. (2) and (4) with the following parameters and initial conditions: and the membership functions are and for the master
Conclusions
This paper is concerned with network-based synchronization of T–S fuzzy chaotic systems. In transmitting signals through a network, we have considered that all communication channels (master-controller, slave-controller, and controller-slave) have asynchronous samplings and external disturbances. As explained earlier, asynchronous samplings can cause two problems: one is asynchronous sampling between sampled-data of the master and slave system in the controller, the other is asynchronous
Acknowledgments
The work of T.H. Lee was supported by the Brain Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, CT and Future Planning (Grant no. NRF-2017M3C7A1044815). The work of J.H. Park was supported by Basic Science Research Programs through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant no. NRF-2017R1A2B2004671).
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